Non-homogeneous slope seismic displacement probability analysis method and system based on finite difference

Abstract

This invention discloses a method and system for probabilistic analysis of seismic displacement of heterogeneous slopes based on finite difference, which evaluates the seismic performance of important engineering slopes by generating slope sliding displacement hazard curves. The method and system first determine the statistical quantities of slope soil parameters and select seismic wave records that match the slope site conditions; then, it generates random field samples of soil parameters and iteratively executes stochastic finite difference numerical simulations to calculate the cumulative slope displacement; next, it constructs a slope displacement prediction model using seismic motion parameters and displacement data, and optimizes the seismic motion parameters based on the model's standard deviation; finally, it develops a slope seismic sliding displacement hazard curve within the framework of probabilistic seismic hazard analysis, estimating the displacement value corresponding to the exceedance probability in the target year. This invention can effectively characterize the spatial variability of soil parameters and the uncertainty of seismic loads, and capture the stress-deformation mechanism of slopes under seismic loading, providing an effective approach for the seismic design of highly important engineering slopes.

Description

Technical Field

[0001] This invention belongs to the field of disaster prediction technology, and relates to a method and system for predicting and analyzing permanent displacement of slopes under earthquake action, and particularly to a method and system for probabilistic analysis of seismic displacement of heterogeneous slopes based on finite difference. Background Technology

[0002] Earthquake-induced landslides are characterized by their wide reach, large scale, and high destructiveness, posing a serious threat to the safety of people's lives and property. Permanent slope displacement is a commonly used indicator for evaluating slope seismic stability, making effective prediction of slope seismic displacement of significant engineering importance. Compared to quasi-static slope stability analysis and Newmark sliding displacement analysis, numerical simulation considering stress-deformation mechanisms can provide more accurate slope performance estimates, especially suitable for slope projects of high importance. The probabilistic analysis method for the hazard of slope seismic sliding displacement can consider the influence of various uncertainties and estimate the displacement value corresponding to a specified seismic fortification standard (or return period), and is currently receiving considerable attention both domestically and internationally. However, slope seismic displacement prediction is usually achieved using semi-empirical models based on the Newmark sliding method, whose prediction accuracy generally has a large deviation, often making it unacceptable to designers of important engineering projects.

[0003] In recent years, some studies have incorporated numerical simulation-based displacement prediction models into probabilistic analysis, providing a more accurate approach for evaluating the seismic performance of important slope engineering projects. However, existing numerical simulation-based displacement probability analysis methods typically assume a uniform spatial distribution of soil parameters within the same soil layer, failing to reflect the coupling influence of soil spatial variability and topographic effects on slope dynamic response and slip patterns. Therefore, it is necessary to reasonably characterize soil spatial variability in numerical simulation-based probabilistic analysis of slope seismic slip hazard and to develop a method and system for probabilistic analysis of seismic displacement of heterogeneous slopes. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of existing methods by providing a conceptually clear, easy-to-operate, and highly accurate method and system for probabilistic analysis of seismic displacement on heterogeneous slopes based on finite difference.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] The method for probabilistic analysis of seismic displacement of heterogeneous slopes based on finite difference includes the following steps:

[0007] Step 1: Estimate the statistical parameters of the slope soil based on the field survey data, and select m seismic wave records that match the site conditions of the slope.

[0008] Step 2: Establish a finite difference numerical simulation model for the slope, divide the model into n material elements, extract the coordinates of the center point of each element, and generate N sets of random field samples X of soil parameters using the midpoint method. i , i = 1, 2, ..., N;

[0009] Step 3: Sample X from the random field i Assign the load to the corresponding slope model element, apply the k-th seismic wave as a load at the bottom boundary of the model, and perform finite difference calculation to determine the maximum cumulative displacement D of the slope surface. i,k This step is executed N×m times, i = 1, 2, ..., N, k = 1, 2, ..., m;

[0010] Step 4: Select different types of ground motion parameters IM and calculate the ground motion parameter value IM corresponding to each seismic wave. k For k = 1, 2, ..., m, establish a slope seismic displacement prediction model D = f(IM);

[0011] Step 5: Calculate the model standard deviation σ for all candidate seismic motion parameters. lnD Sort the data and assign σ to the appropriate values. lnD The minimum IM is used as the target ground motion parameter IM T The probabilistic seismic hazard analysis yielded the IM. T The annual average rate density MRD, i.e., the relationship between IM value and frequency;

[0012] Step 6: Combine the annual average rate density (MRD) with the displacement prediction model D = f(IM) T Convolution is performed to generate a slope sliding displacement hazard curve, and the displacement value corresponding to the target year exceedance probability is estimated.

[0013] Furthermore, in step 1, the spatial variability of soil parameters is described using random field theory;

[0014] Let the vector X = [X1, X2, ..., X... n [Refers to a certain soil parameter distributed at n locations in space, where all variables in vector X follow the same marginal distribution, with mean and coefficient of variation μ.] X and COV X The correlation coefficient ρ(X1,X2) between variables can be obtained from the exponential autocorrelation function:

[0015]

[0016] In the formula, The horizontal spacing between the first and second random field units; δ represents the vertical spacing between the first and second random field units. h and δ vThese represent the horizontal and vertical fluctuation ranges, respectively, reflecting the degree of spatial autocorrelation, or non-uniformity, of soil parameters. When sufficient field investigation data is available, the statistical measure μ... X COV X δ h and δ v It can be obtained directly through moment estimation or maximum likelihood estimation; otherwise, it can be obtained by referring to field data from similar projects or literature suggestions.

[0017] Furthermore, in step 2, a finite difference numerical simulation model of the slope is established based on the field survey data. The model is divided into n material elements, the coordinates of the center point of each element are extracted, and the autocorrelation coefficient matrix R of the random field variables is calculated.

[0018]

[0019] Furthermore, in step 2, N sets of random field samples X of soil parameters are generated using the midpoint method. i For i = 1, 2, ..., N, the correlation coefficient matrix R can be decomposed into a lower triangular matrix L using the Choleski decomposition method, which can be expressed as:

[0020] LL T =R

[0021] In the formula, T represents the matrix transpose. Therefore, a set of random field samples X can be simulated as:

[0022] X = F -1 [Φ(LU)]

[0023] In the formula, U is an independent standard normal sample vector of dimension n×1; Φ(·) is the cumulative distribution function of the standard normal distribution variable; F -1 (·) is the inverse function of the marginal cumulative distribution function of the soil parameters. Repeating the above equation N times will yield all random field samples, which can be used for subsequent probability calculations.

[0024] Furthermore, in step 3, the viscous boundary, i.e., the static boundary, is used to simulate the semi-infinite foundation conditions at the bottom, and the acceleration time history a of the k-th seismic wave is obtained. k (t) is converted to stress time history τ k (t) is then applied to the viscous boundary, and the transformation formula is:

[0025]

[0026] In the formula, G max ρ and v are the initial shear modulus and density of the material at the bottom of the model, respectively; k(t) represents the velocity time history after integrating the seismic wave acceleration. A dynamic finite-difference simulation is performed, recording the cumulative displacement time histories at several monitoring points on the slope surface. The maximum cumulative displacement value at the end of the time history is taken as D. i,k .

[0027] Furthermore, the specific implementation of step 4 includes the following sub-steps:

[0028] Step 4.1: Calculate the m sets of ground motion parameter values ​​IM k Since each set of ground motion parameter values ​​corresponds to N sets of random field samples, each type of IM is... k The value is copied N times to get IM i,k k = 1, 2, ..., m;

[0029] Step 4.2: Utilize N×m sets of data D i,k and IM i,k The displacement prediction model D = f(IM) corresponding to various seismic motion parameters is obtained through regression. The specific form of the selected model is as follows:

[0030]

[0031] In the formula, ln represents the natural logarithm symbol; is the predicted value of the slope sliding displacement, in cm; IM represents any of the selected seismic motion parameters; a0 and a1 are regression coefficients; ε is a standard normal distribution variable; σ lnD σ represents the model standard deviation, reflecting prediction uncertainty. lnD The smaller the value, the stronger the model's effectiveness in predicting slope displacement.

[0032] Furthermore, in step 5, MRD(z) can be considered as the annual mean exceedance probability λ of the seismic motion parameter. IM The rate of change of (z) with respect to the parameter value z is obtained in the following way:

[0033]

[0034] In the formula, λ0 is the seismic focal activity rate; f M (m) and f R (r) are the probability density functions of magnitude M and fault distance R, respectively; f IM (z|m,r) is the probability density function of IM at M=m and R=r, which can be expressed as:

[0035]

[0036] In the formula, μ lnIM and σ lnIMIt is derived from a certain seismic motion parameter prediction equation, GMPE. GMPE is generally obtained through regression analysis of a large number of measured seismic motion records, and its essence is μ. lnIM or σ lnIM Functions relating to seismic load parameters.

[0037] Furthermore, in step 6, the sliding displacement hazard curve describes the annual average exceedance rate λ of the displacement. D (d) The relationship between λ and the threshold d, where λ D (d) can be calculated as:

[0038] λ D (d)=∫P[D>dIM T =z]MRD IM (z)dz

[0039] In the formula, P[D>d|IM T =z] is IM T The probability that the displacement D exceeds the threshold d when z = z can be obtained by the following formula:

[0040]

[0041] In the formula, Φ(·) is the cumulative distribution function of the standard normal distribution variable. Substituting different values ​​of d into the above formula, the corresponding λ is calculated. D (d) can then be used to plot the slope sliding displacement hazard curve, which can be used to estimate the slope displacement value corresponding to the probability level (recurrence period) of any target exceeding the limit.

[0042] This invention also provides a seismic displacement probability analysis system for heterogeneous slopes based on finite difference, characterized by comprising the following modules:

[0043] Module 1: Estimating statistical parameters of slope soil and selecting seismic records that match the site conditions of the slope.

[0044] Module 2: Construct a finite difference numerical simulation model of the slope and generate random field samples of soil parameters;

[0045] Module 3, Automated calculation of seismic sliding displacement of heterogeneous slopes under a series of random field samples of soil parameters and seismic acceleration time history;

[0046] Module 4: Construct a slope seismic displacement prediction model using seismic motion parameters as input;

[0047] Module 5 enables the comparison and selection of seismic ground motion parameters and the probabilistic seismic hazard analysis based on the optimal seismic ground motion parameters;

[0048] Module 6 generates the slope seismic sliding displacement hazard curve and estimates the displacement value corresponding to the target year exceedance probability.

[0049] Furthermore, the slope seismic displacement prediction model selected by module 4 is specifically in the following form:

[0050]

[0051] In the formula, ln represents the natural logarithm symbol; is the predicted value of the slope sliding displacement, in cm; IM represents any of the selected seismic motion parameters; a0 and a1 are regression coefficients; ε is a standard normal distribution variable; σ lnD σ represents the model standard deviation, reflecting prediction uncertainty. lnD The smaller the value, the stronger the model's effectiveness in predicting slope displacement.

[0052] Compared with the prior art, this application has the following beneficial effects:

[0053] (1) This invention provides a method for seismic displacement probability analysis of heterogeneous slopes, which is applicable to different seismic conditions and slope conditions.

[0054] (2) The method of the present invention is based on finite difference numerical simulation, which can capture the stress deformation mechanism of slope under seismic action more accurately than the traditional Newmark slider analysis.

[0055] (3) The method of the present invention realizes the spatial variability of soil parameters based on random field theory and the uncertainty characterization of seismic load based on probabilistic seismic hazard analysis. Attached Figure Description

[0056] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0057] Figure 1 This is a schematic diagram of the method flow according to an embodiment of the present invention;

[0058] Figure 2 This is a selected seismic wave spectral acceleration distribution map in an embodiment of the present invention;

[0059] Figure 3 The diagram shows the slope model and the results of the deterministic quasi-static slope stability analysis in this embodiment of the invention.

[0060] Figure 4 This is a typical implementation cloud diagram of the friction angle random field in the embodiments of the present invention;

[0061] Figure 5This is a distribution diagram showing the correlation between slope displacement and seismic motion parameters in an embodiment of the present invention;

[0062] Figure 6 This is a comparison chart of the standard deviations of the displacement models in the embodiments of the present invention;

[0063] Figure 7 This is a seismic hazard curve based on spectral intensity SI in an embodiment of the present invention;

[0064] Figure 8 This is the annual average rate-density curve of the spectral intensity SI in an embodiment of the present invention;

[0065] Figure 9 This is a slope seismic sliding displacement hazard curve in an embodiment of the present invention. Detailed Implementation

[0066] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0067] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0068] The present invention will now be described in detail with reference to the accompanying drawings. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0069] Example: Please see Figure 1 The present invention provides a method for probabilistic analysis of seismic displacement of heterogeneous slopes based on finite difference, comprising the following steps:

[0070] Step 1: Estimate the statistical parameters of the slope soil (such as mean, coefficient of variation, and fluctuation range) based on the field survey data, and select m seismic wave records that match the site conditions of the slope.

[0071] Soil is subject to the combined effects of various geological, environmental, and physicochemical processes, and in-situ soil parameters often vary along both vertical and horizontal directions, exhibiting spatial variability. Over the past few decades, random field theory has been considered the most effective tool for describing the spatial variability of soil parameters. Let vector X = [X1, X2, ..., X...] n[Refers to a certain soil parameter (such as the friction angle φ) distributed at n locations in space, where all variables in the vector X follow the same marginal distribution (such as a log-normal distribution), with mean and coefficient of variation μ. X and COV X The correlation coefficient ρ(X1,X2) between variables can be obtained from the exponential autocorrelation function:

[0072]

[0073] In the formula, The horizontal spacing between the first and second random field units; δ represents the vertical spacing between the first and second random field units. h and δ v These represent the horizontal and vertical fluctuation ranges, respectively, reflecting the degree of spatial autocorrelation (non-homogeneity) of soil parameters. When sufficient field investigation data (such as static cone penetration test (CPT) data is available, the statistic μ... X COV X δ h and δ v It can be obtained directly through moment estimation or maximum likelihood estimation; otherwise, it can be obtained by referring to field data from similar projects or literature suggestions.

[0074] In this embodiment, seismic waves are selected from the NGA-West2 global ground motion database, and m seismic wave records that match the slope site conditions (such as soil or rock sites) are chosen. The selection principle is to cover a wide range of earthquake magnitudes and fault distances, so as to take into account different characteristics of ground motion (such as amplitude and frequency) as much as possible, so that the subsequent slope displacement prediction model can be well applied to different earthquake scenarios.

[0075] Step 2: Establish a finite difference numerical simulation model for the slope, divide the model into n material elements, extract the coordinates of the center point of each element, and generate N sets of random field samples X of soil parameters using the midpoint method. i (i = 1, 2, ..., N);

[0076] Step 2.1: Based on the field survey data, establish a finite difference numerical simulation model of the slope, divide the model into n material elements, extract the coordinates of the center point of each element, and calculate the autocorrelation coefficient matrix R of the random field variables.

[0077]

[0078] In this embodiment, the finite difference software FLAC is used for slope dynamic analysis. The cyclic stress-strain characteristics of the soil are described by the Mohr-Coulomb plasticity criterion in FLAC combined with the hysteresis damping model Sig4. The uncertain soil parameters considered are cohesion (c), friction angle (φ), and initial shear modulus (G). max This criterion can explain the nonlinear behavior of soil in the elastic stage of dynamic response and can also simulate potential irreversible strain accumulation, where the parameters of model Sig4 are calibrated by a nonlinear soil dynamic property model in the literature. To minimize the influence of wave reflection at the model boundary on the dynamic response analysis, free field boundary conditions are applied on both sides of the model, and the lateral boundaries are set at a sufficiently far distance from the slope.

[0079] Step 2.2: Generate N sets of random field samples X of soil parameters using the midpoint method. i (i = 1, 2, ..., N). Using the Choleski decomposition, the correlation matrix R is decomposed into a lower triangular matrix L, which can be expressed as:

[0080] LL T =R

[0081] In the formula, T represents the matrix transpose. Therefore, a set of random field samples X can be simulated as:

[0082] X = F -1 [Φ(LU)]

[0083] In the formula, U is an independent standard normal sample vector of dimension n×1; Φ(·) is the cumulative distribution function of the standard normal distribution variable; F -1 (·) is the inverse function of the marginal cumulative distribution function of the soil parameters. Repeating the above equation N times yields all random field samples, which can be used for subsequent probability calculations.

[0084] Step 3: Sample X from the random field i Assign the load to the corresponding slope model element, apply the k-th seismic wave as a load at the bottom boundary of the model, and perform finite difference calculation to determine the maximum cumulative displacement D of the slope surface. i,k .

[0085] Using viscous boundaries (static boundaries) to simulate the semi-infinite foundation conditions at the bottom, the acceleration time history a of the k-th seismic wave is obtained. k (t) is converted to stress time history τ k (t) is then applied to the viscous boundary, and the transformation formula is:

[0086]

[0087] In the formula, G max ρ and v are the initial shear modulus and density of the material at the bottom of the model, respectively;k (t) represents the velocity-time history after integrating the seismic wave acceleration. A dynamic finite-difference simulation is performed, recording the cumulative displacement-time history at several monitoring points on the slope surface. The maximum cumulative displacement value at the end of the time history is taken as D. i,k Step 3 is executed N×m times, i.e., i = 1, 2, ..., N, k = 1, 2, ..., m, resulting in a total of N×m slope displacement values.

[0088] Step 4: Select different types of ground motion parameters (IM) and calculate the IM corresponding to each seismic wave. k Values ​​(k=1,2,…,m) are used to establish a slope seismic displacement prediction model D=f(IM).

[0089] Step 4.1: Taking three ground motion parameters as an example, including peak ground acceleration (PGA), peak ground velocity (PGV), and Arias intensity (IL). A It is necessary to write a custom program or use commercial software (such as SeismoSignal) to calculate m sets of ground motion parameter values ​​IM. k , namely PGA k PGV k and I A,k (k = 1, 2, ..., m). Since each set of seismic motion parameter values ​​corresponds to N sets of random field samples, each IM... k The value is copied N times to get IM i,k , namely PGA i,k PGV i,k and I A,i,k (k=1,2,…,m, i=1,2,…,N).

[0090] Step 4.2: Utilize N×m sets of data D i,k and IM i,k Regression yields the displacement prediction model D = f(IM) corresponding to various seismic motion parameters. To ensure simplicity in engineering practice, the selected model takes the following specific form:

[0091]

[0092] In the formula, ln represents the natural logarithm symbol; σ represents the predicted slope sliding displacement in cm; IM represents any of the selected seismic motion parameters; a0 and a1 are regression coefficients; ε is a standard normally distributed variable (mean 0, standard deviation 1); σ lnD σ represents the model standard deviation, reflecting prediction uncertainty. lnD The smaller the value, the stronger the model's effectiveness in predicting slope displacement.

[0093] Step 5: For all candidate ground motion parameters, calculate σ. lnD Sort the data and assign σ to the appropriate values.lnD The minimum IM is used as the target ground motion parameter IM T Probabilistic seismic hazard analysis (PSHA) was performed to obtain the IM. T The annual average rate density (MRD) is the relationship between IM value and frequency.

[0094] PSHA can be summarized into four basic steps: delineating potential source areas, establishing earthquake recurrence models for potential source areas, describing the probability distribution of ground motion parameters, and estimating the seismic hazard of the site (e.g., plotting the annual exceedance probability curve of ground motion parameters). After decades of theoretical research and practice, PSHA has become the basis for the compilation of ground motion parameter zoning maps in most seismic design codes, such as my country's "Code for Seismic Design of Buildings" (GB50011-2010). MRD(z) can be considered as the annual mean exceedance probability λ of ground motion parameters. IM The rate of change of (z) with respect to the parameter value z is obtained in the following way:

[0095]

[0096] In the formula, λ0 is the seismic focal activity rate; f M (m) and f R (r) are the probability density functions of magnitude (M) and fault distance (R), respectively; f IM (z|m,r) is the probability density function of IM at M=m and R=r, which can be expressed as:

[0097]

[0098] In the formula, μ lnIM and σ lnIM It is obtained from a certain ground motion parameter prediction equation (GMPE). GMPE is generally obtained through regression analysis of a large number of measured ground motion records, and its essence is μ. lnIM or σ lnIM Functions relating to seismic load parameters (such as M and R).

[0099] Step 6: Combine the MRD obtained in Step 5 with the displacement prediction model D = f(IM) obtained in Step 4. T Convolution is performed to generate a slope sliding displacement hazard curve, estimating the displacement value corresponding to the target year's exceedance probability. The sliding displacement hazard curve describes the annual average exceedance rate λ. D (d) The relationship between λ and the threshold d, where λ D (d) can be calculated as:

[0100] λ D (d)=∫P[D>dIM T =z]MRD IM (z)dz

[0101] In the formula, P[D>d|IM T =z] is IM T The probability that the displacement D exceeds the threshold d when z = z can be obtained by the following formula:

[0102]

[0103] In the formula, Φ(·) is the cumulative distribution function of the standard normal distribution variable. Substituting different values ​​of d into the above formula calculates the corresponding λ. D (d) can then be used to plot the slope sliding displacement hazard curve, which can be used to estimate the slope displacement value corresponding to the probability level (recurrence period) of any target exceeding the limit.

[0104] The present invention will now be further described with reference to specific embodiments.

[0105] I. Introduction to Slope and Seismic Source Models

[0106] Based on the method proposed in this invention, this example will perform a probabilistic analysis of the seismic sliding displacement of a simplified slope model. The slope height and slope angle are 20m and 30°, respectively, and the thicknesses of the clay layer and bedrock layer are 40m and 10m, respectively. The clay's cohesion c, friction angle φ, and initial shear modulus G are also considered. max The reference values ​​for specific gravity γ were taken as 55 kPa, 14°, 180 MPa, and 19.62 kN / m, respectively. 3 G of bedrock max The reference values ​​for γ are taken as 1472 MPa and 22.56 kN / m, respectively. 3 Using a point source model as the potential source, the source activity rate is set to λ0 = 0.1, and a total of 12 magnitude scenarios are considered, ranging from 5 to 8 (with an increment step of 0.25). It is assumed that the slope is located in a rocky site 10 kilometers away from the source.

[0107] II. Specific Implementation Process

[0108] Step 1, combine the c, φ, and G of the clay. max Treated as uncertain parameters, statistical quantities (such as mean, coefficient of variation, and fluctuation range) of slope soil parameters are estimated based on survey data or engineering data, as shown in Table 1. Additionally, 83 seismic waves were selected from the NGA-West2 seismic ground motion database, corresponding to magnitudes (M) ranging from 5.6 to 7.4, fault distances (R) ranging from 0.9 to 83 km, and equivalent shear wave velocities (V) at 30 m near the surface. s30 (greater than 600m / s) Figure 2 The response spectra of the selected seismic waves were plotted, with peak ground acceleration (PGA) ranging from 0.034g to 1.406g. The figures show that the characteristics (such as amplitude and period) of different ground motions vary considerably, allowing for a full consideration of ground motion variability.

[0109] Table 1. Slope Soil and Rock Mass Parameters and Statistics

[0110]

[0111]

[0112] Step 2: Establish a finite difference numerical simulation model for the slope. Divide the model into 12066 material elements, extract the coordinates of the center point of each element, and calibrate the parameters of Sig4 using the semi-empirical clay model proposed by Darendeli (2001) to characterize the variation trend of soil shear modulus and damping ratio with shear strain. In addition, a small amount of Rayleigh damping (0.2%) proportional to stiffness is set for the soil material to remove high-frequency noise, and Rayleigh damping (0.5%) proportional to mass and stiffness is set for the elastic bedrock material. Figure 3 The geometry of the slope model and the shear zone (maximum shear strain increment contour map) generated by the quasi-static strength reduction method are shown, indicating that the slope is prone to deep sliding under seismic loading, with a large sliding volume. Subsequently, 100 sets of c, φ, and G values ​​are generated using the midpoint method. max random field sample X i (i = 1, 2, ..., 100).

[0113] Step 3, take the random field sample X i Assign the material element to the corresponding slope model and apply the k-th seismic wave as a dynamic load at the bottom boundary of the model. Figure 4 A set of random field implementations for φ in the slope model is presented, demonstrating that the random field model effectively describes the spatial non-uniformity of soil parameters. A series of displacement monitoring points are arranged along the slope surface, and the maximum cumulative displacement D of the slope surface is calculated using finite difference methods. i,k (i=1,2,…,100; k=1,2,…,83).

[0114] Step 4: Select eight common ground motion parameters (IM), including peak ground acceleration (PGA), peak ground velocity (PGV), and Arias intensity (I). A ), Cumulative Absolute Velocity (CAV), Average Period (T) m ), spectral intensity (SI), acceleration spectral intensity (ASI), 70% energy duration (Ds) 5-75 Calculate the IM corresponding to each seismic wave. k Values ​​(k = 1, 2, ..., 83), using displacement and IM k Data regression yields the displacement prediction model D = f(IM) and its standard deviation σ corresponding to various seismic motion parameters. lnDThe regression coefficients a0 and a1 for each model are shown in Table 2. The distribution of seismic motion parameters and displacements, as well as the fitting of the displacement models, are shown in [Table 2]. Figure 5 As can be seen from the graph, the displacement data has a large dispersion, except for T. m and Ds 5-75 In addition, displacement models based on other ground motion parameters can reasonably predict slope displacement, but parameter SI obviously produces the best fitting effect.

[0115] Table 2 Regression coefficients of the displacement prediction model

[0116]

[0117] Step 5, for all candidate seismic motion parameters, the corresponding σ lnD Sort the data, and the sorting result is as follows: Figure 6 As shown, σ corresponds to SI. lnD Since SI is the minimum, it is used as the target ground motion parameter. Considering the aforementioned point source model, probabilistic seismic hazard analysis (PSHA) is performed to obtain the annual average exceedance probability curve and annual average rate-density (MRD) curve of SI against different thresholds. Figure 7 and Figure 8 The figure shows that ln(SI) approximately follows a normal distribution, consistent with the assumptions made in calculating MRD.

[0118] Step 6: Convolve the MRD with the displacement prediction model lnD = -5.47 + 2.05ln(SI) + 0.65ε to generate the slope sliding displacement hazard curve corresponding to SI. For comparison, the seismic motion parameter Ds that leads to the worst displacement prediction performance is also derived. 5-75 The corresponding displacement hazard curve. Figure 9 This demonstrates the optimal parameter SI and the worst parameter Ds. 5-75 The displacement hazard curve is generated, and it can be seen that Ds is selected. 5-75 This would significantly overestimate slope seismic displacement, leading to overly conservative and uneconomical slope seismic design schemes. If the design life is 475 years, the maximum possible displacement of the slope within this lifespan can be calculated as 48 cm based on the SI slope displacement hazard curve. Engineers need to consider the specific seismic design standards for the slope to determine whether to take measures such as slope protection and reinforcement.

[0119] This embodiment also provides a seismic displacement probability analysis system for heterogeneous slopes based on finite difference, characterized by including the following modules:

[0120] Module 1: Estimating statistical parameters of slope soil and selecting seismic records that match the site conditions of the slope.

[0121] Module 2: Construct a finite difference numerical simulation model of the slope and generate random field samples of soil parameters;

[0122] Module 3, Automated calculation of seismic sliding displacement of heterogeneous slopes under a series of random field samples of soil parameters and seismic acceleration time history;

[0123] Module 4: Construct a slope seismic displacement prediction model using seismic motion parameters as input;

[0124] Module 5 enables the comparison and selection of seismic ground motion parameters and the probabilistic seismic hazard analysis based on the optimal seismic ground motion parameters;

[0125] Module 6 generates the slope seismic sliding displacement hazard curve and estimates the displacement value corresponding to the target year exceedance probability.

[0126] The above description of the preferred embodiments is quite detailed, but it should not be considered as a limitation on the scope of protection of this invention. Those skilled in the art, under the guidance of this invention, can make substitutions or modifications without departing from the scope of protection of the claims of this invention, and all such substitutions or modifications fall within the scope of protection of this invention. The scope of protection of this invention should be determined by the appended claims.

Claims

1. A finite-difference-based probabilistic analysis method for seismic displacement of a heterogeneous slope, characterized by, Comprising the following steps: Step 1: Estimate the statistical quantity of the soil parameters of the slope according to the site investigation data, and select the same matching conditions as the slope site m strip seismic record; Step 2: Establish the finite difference numerical simulation model of the slope, divide the model into n material units, extract the coordinates of the center points of each unit, and generate N a set of soil parameter random field samples X i , i = 1, 2, …, N ; Step 3: Random field sample X is given to the corresponding slope model unit, and the first i boundary at the bottom of the model is applied k The maximum cumulative displacement of the slope surface is calculated by performing finite difference calculation on the seismic wave as load D i,k , loop N X m times to execute this step, that is i =1, 2,…, N , k = 1, 2,…, m ; Step 4: Selecting different types of ground motion parameters IM , calculate the ground motion parameter value corresponding to each seismic wave IM k , k = 1,2,…, m , establish a slope seismic displacement prediction model D = f ( IM ); Comprising the following sub-steps: Step 4.1: Calculate m Group ground motion parameter values IM k Since each group of ground motion parameter values corresponds N to a group of random field samples, each of the IM k values is replicated N times to obtain IM i,k , k = 1, 2, …, m , i = 1, 2, …, N ; Step 4.2: Utilizing N × m Group data D i,k and IM i,k Regression obtained displacement prediction model corresponding to each type of ground motion parameter D = f ( IM ), the selected model specific form is: ln represents natural logarithm symbol; is the predicted value of slope sliding displacement, with unit of cm; IM represents any one of the selected ground motion parameters; a 0 and a 1 is the regression coefficient; ε is the standard normal distribution variable; σ lnD is the model standard deviation, reflecting the prediction uncertainty, σ lnD The smaller the value is, the stronger the effectiveness of the model in predicting the slope displacement is. Step 5: Model standard deviation for all alternative ground motion parameters σ lnD Ranking, for each target ground motion parameter σ lnD the smallest IM IM T Performing a probabilistic seismic hazard analysis yields IM T the annual mean rate density MRD i.e. IM the relationship between the value and the frequency;​ Step 6: Convolve the annual mean rate density MRD with the displacement prediction model D = f ( IM T ) to generate a slope sliding displacement hazard curve that estimates the displacement value corresponding to the target annual exceedance probability.

2. The finite-difference based probabilistic analysis method of seismic displacement of a heterogeneous slope according to claim 1, characterized in that: In step 1, the soil parameter spatial variability is described by using the random field theory; Let X = [ X 1, X 2,…, X n ] denote a certain soil parameter distributed in space n at L locations, each variable in vector X follows the same marginal distribution with mean μ X and coefficient of variation COV X , and the correlation coefficient between variables can be obtained from an exponential autocorrelation function: wherein, is the horizontal distance between the 1st and 2nd random field units; is the vertical distance between the 1st and 2nd random field units; δ h and δ v are the horizontal and vertical fluctuation ranges, respectively, reflecting the degree of spatial correlation, i.e. non-uniformity, of the soil parameters. When the field investigation data are sufficient, the statistical quantities μ X , COV X , δ h and δ v can be directly obtained by the moment estimation or maximum likelihood estimation, otherwise, the values can be taken from the field data or literature recommendations of similar projects.

3. The finite-difference based probabilistic analysis method of seismic displacement of a heterogeneous slope according to claim 1, characterized in that: In step 2, a finite difference numerical simulation model of the slope is established according to the site survey data, the model is divided into n material units, the center point coordinates of each unit are extracted, and a self-correlation coefficient matrix R of the random field variable is calculated. 。 4. The finite-difference based probabilistic analysis method of seismic displacement of a heterogeneous slope according to claim 3, characterized in that: In step 2, the midpoint method is used to generate N Group soil body parameter random sample X i , i = 1, 2, …, N By Cholesky decomposition, the correlation coefficient matrix R is decomposed into a lower triangular matrix L, which can be expressed as: In the formula, T represents the matrix transpose, so a set of random field samples X can be simulated as: where U is a vector of independent standard normal samples of dimension n Φ(·) is the cumulative distribution function of the standard normal distribution variable; is the inverse function of the marginal cumulative distribution function of the soil parameters, and the above equation is repeated N times to obtain all the random field samples for subsequent probabilistic calculations.

5. The finite-difference based probabilistic analysis method of seismic displacement of a heterogeneous slope according to claim 1, wherein, In step 3, the bottom half-infinite foundation condition is simulated by using a viscous boundary, i.e., a static boundary, and the first k acceleration time history of the seismic wave a k t converted to a stress time history τ k t applied on the viscous boundary, the conversion formula is:​​ wherein, G max and ρ are the initial shear modulus and density of the model base material, respectively; v k t is the velocity time history after the integration of the seismic wave acceleration, a one-dimensional dynamic finite difference simulation is performed, the cumulative displacement time history of several monitoring points on the slope surface is recorded, and the maximum cumulative displacement value at the end of the time history is taken as D i,k .​ 6. The finite-difference based probabilistic analysis method of seismic displacement of a heterogeneous slope according to claim 1, wherein, In step 5, MRD z may be considered as the annual average exceedance probability of the ground motion parameter λ IM z with the rate of change of the parameter value z is obtained by​​ In the formula, λ 0 represents the earthquake focal activity rate; f M ( m )and f R ( r The magnitudes are respectively M and fault distance R The probability density function; f IM ( z | m , r )for IM exist M = m and R = r The probability density function at time t can be expressed as: In the formula, μ lnIM and σ lnIM is obtained from a certain ground motion parameter prediction equation GMPE, which is generally obtained by regression analysis on a large number of measured ground motion records, and its essence is μ lnIM or σ lnIM a function of the seismic operating parameter.

7. The finite-difference based probabilistic analysis method of seismic displacement of a heterogeneous slope according to claim 1, wherein, In step 6, the sliding displacement hazard curve describes the annual mean exceedance rate of displacement λ D ( d ) as a function of the threshold d , where λ D ( d ) can be calculated as: wherein P [ D > d | IM T = z ] is IM T = z the probability that the displacement D exceeds the threshold value d can be obtained from the equation: where Φ(·) is the cumulative distribution function of the standard normal variable, and different values of d are substituted into the above equation to calculate the corresponding λ D ( d ), which can be used to estimate the slope displacement value corresponding to any target exceedance probability level.

8. A system for performing the steps of the finite-difference based probabilistic analysis method of seismic displacement of heterogeneous slopes according to any one of claims 1 to 7, characterized in that, Comprising the following modules: Module 1, estimating the statistical quantity of the slope soil parameters and selecting the seismic wave record matched with the slope site condition; Module 2, constructing the finite difference numerical simulation model of the slope and generating the soil parameter random field sample; Module 3, automatically calculating the seismic sliding displacement of the heterogeneous slope under a series of soil parameter random field samples and seismic acceleration time history; Module 4, constructing the slope seismic displacement prediction model taking the ground motion parameters as the input; Module 5, realizing the comparison and selection of the ground motion parameters and the probability seismic risk analysis based on the optimal ground motion parameters; Module 6, generating the slope seismic sliding displacement risk curve and estimating the displacement value corresponding to the target year exceedance probability.

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